{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# $$.9\\overline{9} = 1?$$\n",
    "\n",
    "# <center>Why every proof you have seen is WRONG</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Don't worry, **it is true**. Just the \"proofs\" you've seen are wrong."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# First, why the common \"proofs\" are wrong\n",
    "# Then, an actual proof"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Algebraic \"proofs\" are ALL wrong\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "x &:= .9\\overline{9} &&&1/3 &= .3\\overline{3} \\\\\n",
    "10x &= 9.9\\overline{9} &\\text {or}&& 3\\cdot (1/3) &= 3\\cdot(.3\\overline{3})\\\\\n",
    "9x &= 9 &&& 1 &= .9\\overline{9}\\\\\n",
    "x &= 1\n",
    "\\end{align}\n",
    "$$\n",
    "All algebraic proofs are wrong because there are algebras where $.9\\overline{9} \\neq 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "x &:= \\bf\\color{red}{.9\\overline{9}} &&&1/3 &= \\bf\\color{red}{.3\\overline{3}} \\\\\n",
    "10x &= 9.9\\overline{9} &\\text {or}&& 3\\cdot (1/3) &= 3\\cdot(.3\\overline{3})\\\\\n",
    "9x &= 9 &&& 1 &= .9\\overline{9}\\\\\n",
    "x &= 1\n",
    "\\end{align}\n",
    "$$\n",
    "You have already assumed part of the conclusion, that $\\color{red}{.9\\overline{9}}$ or $\\color{red}{.3\\overline{3}}$ even *exists*."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "x &:= \\bf\\color{red}{\\overline{9}.} \\\\\n",
    "10x &= \\overline{9}0.  \\\\\n",
    "10x+9 &= x\\\\\n",
    "9x &= -9\\\\\n",
    "x &= -1\n",
    "\\end{align}\n",
    "$$\n",
    "So do you believe that $...99999999.0 = -1$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$$\\color{green}{\n",
    "\\begin{align}\n",
    "\\color{black}{x} &\\color{black}{:= .9\\overline{9}} &&&1/3 &= .3\\overline{3} \\\\\n",
    "10x &= 9.9\\overline{9} &\\color{black}{\\text {or}}&& 3\\cdot (1/3) &= 3\\cdot(.3\\overline{3})\\\\\n",
    "9x &= 9 &&& 1 &= .9\\overline{9}\\\\\n",
    "x &= 1\n",
    "\\end{align}}\n",
    "$$\n",
    "Okay, suppose you are willing to believe $.9\\overline{9}$ is a real number, what else is wrong?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "&&x &:= .9\\overline{9} &&&&&1/3 &= .3\\overline{3} \\\\\n",
    "\\implies&& 10x &= 9.9\\overline{9} &\\text {or}&&&\\implies& 3\\cdot (1/3) &= 3\\cdot(.3\\overline{3})\\\\\n",
    "\\implies&&9x &= 9 &&&&\\implies& 1 &= .9\\overline{9}\\\\\n",
    "\\implies&& x &= 1\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "&&x &:= .9\\overline{9} &&&&&1/3 &= .3\\overline{3} \\\\\n",
    "\\color{red}{\\implies}&& 10x &= 9.9\\overline{9} &\\text {or}&&&\\implies& 3\\cdot (1/3) &= 3\\cdot(.3\\overline{3})\\\\\n",
    "\\implies&&9x &= 9 &&&&\\bf\\color{red}{\\implies}& 1 &= .9\\overline{9}\\\\\n",
    "\\implies&& x &= 1\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "*Why* does $10 \\cdot (.9\\overline{9}) = 9.9\\overline{9}$?\n",
    "*Why* does $3 \\cdot (.3\\overline{3}) = .9\\overline{9}$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "x &:= .9\\overline{9} &&&1/3 &\\color{red}{=} .3\\overline{3} \\\\\n",
    "10x &= 9.9\\overline{9} &\\text {or}&& 3\\cdot (1/3) &= 3\\cdot(.3\\overline{3})\\\\\n",
    "9x &= 9 &&& 1 &= .9\\overline{9}\\\\\n",
    "x &= 1\n",
    "\\end{align}\n",
    "$$\n",
    "Isn't $1/3 \\color{red}{=} .3\\overline{3}$ just as mysterious? Why believe it?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# The real source of confusion: definition\n",
    "\n",
    "What does $.9\\overline{9}$ actually mean?\n",
    "\n",
    "It is **NOT** a mystical infinite sequence of $9$'s after a decimal point."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The definition of $.9\\overline{9}$:\n",
    "$$\n",
    "\\begin{align}\n",
    "x_1 &:= .9,\\\\ \n",
    "x_2 &:= .99,\\\\\n",
    "x_3 &:= .999, \\ldots \\\\\n",
    "x_n &:= \\sum_{i=1}^n 9 \\cdot 10^{-i} = \\underbrace{.9999}_{n}.\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Then $.9\\overline{9}$ is **defined** to be the number that the sequence $$\n",
    "(x_n)_n = (.9, .99, .999, \\cdots)\n",
    "$$ approaches, **if it exists**. That is, it is by definition the limit of the sequence."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "For any repeating decimal, we use the same process to define it.\n",
    "\n",
    "E.g. for $2.64\\overline{51}$ you would use the sequence \n",
    "$$\n",
    "(2.645, 2.6451, 2.64515, 2.645151, 2.6451515, \\cdots).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# So what is $.9\\overline{9} = 1$ even claiming?\n",
    "\n",
    "\n",
    "It is simply claiming that $$(.9, .99, .999, \\cdots)$$ does approach a real number, and that number is $1$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "scrolled": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "N = np.arange(1, 10)\n",
    "X = np.cumsum(9 * np.power(10.,-N))\n",
    "\n",
    "plt.axhline(1.0, color=\"grey\", linestyle=\"dotted\")\n",
    "plt.plot(N, X, marker=\"x\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Proof (trivial)\n",
    "\n",
    "Compute\n",
    "$$\n",
    "|x_n - 1| = |  \\underbrace{.9999}_{n} - 1| = 10^{-n}.\n",
    "$$\n",
    "For any tolerance $\\epsilon>0$, for any $n > 1/\\epsilon$, we have\n",
    "$$\n",
    "|x_n - 1| = 10^{-n} < 1/n < \\epsilon. \n",
    "$$\n",
    "This formally shows that $x_n$ approaches 1, but you already knew that."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Honorable mentions\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Thanks!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "like()\n",
    "comment()\n",
    "subscribe()\n",
    "while True:\n",
    "    try:\n",
    "        tell_friends()\n",
    "    except KeyError:\n",
    "        tell_family()\n",
    "    await asyncio.sleep(60*60*24)"
   ]
  }
 ],
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    "version": 3
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   "pygments_lexer": "ipython3",
   "version": "3.9.0"
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     "varRefreshCmd": "print(var_dic_list())"
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     "delete_cmd_prefix": "rm(",
     "library": "var_list.r",
     "varRefreshCmd": "cat(var_dic_list()) "
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   "types_to_exclude": [
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